Monotone and Concave Positive Solutions to a Boundary Value Problem for Higher-Order Fractional Differential Equation
We consider boundary value problem for nonlinear fractional differential equation , where denotes the Caputo fractional derivative. By using fixed point theorem, we obtain some new results for the existence and multiplicity of solutions to a higher-order fractional boundary value problem. The interesting point lies in the fact that the solutions here are positive, monotone, and concave.
In this paper, we deal with the following boundary value problem for higher-order fractional differential equation: where denotes the Caputo fractional derivative and is a real function. By using fixed point theorem, some sufficient conditions for existence and multiplicity of solutions to the above boundary value problem are obtained. Moreover, we will show that the solutions obtained here are positive, monotone, and concave.
Fractional differential equations are valuable tools in the modelling of many phenomena in various fields of science and engineering [1–5]. Due to their applications, fractional differential equations have gained considerable attentions and there has been a significant development in the study of existence of solutions, and positive solutions to boundary value problems for fractional differential equations (e.g., [6–9] and references therein).
Some papers are devoted to study the existence of solutions for higher-order fractional boundary value problem. Salem  investigated the existence of pseudosolutions for the nonlinear -point boundary value problem of fractional type Zhang  considered the existence of positive solutions to the singular boundary value problem for fractional differential equations where is the Riemann-Liouville fractional derivative of order . In another paper, Zhang  studied the existence, multiplicity, and nonexistence of positive solutions for the following higher-order fractional boundary value problem: where is the Caputo fractional derivative of order .
It seems that the authors of the papers only studied the existence of the solutions or positive solutions. No one consider the qualities of the solutions for boundary value problems of fractional differential equation. Motivated by all the above works, the aim of this paper is to study the monotone, concave, and positive solutions of a fractional differential equation.
The rest of the paper is organized as follows. In Section 2, we will introduce some lemmas and definitions which will be used later. In Section 3, the existence and multiplicity of positive solutions for the boundary value problem (1.1) will be discussed. In Section 4, examples are given to check our results.
2. Basic Definitions and Preliminaries
In this section, we introduce some necessary definitions and lemmas, which will be used in the proofs of our main results.
Definition 2.1 (see ). The integral where is called the Riemann-Liouville fractional integral of order .
Definition 2.2 (see ). The Caputo fractional derivative for a function can be written as where , denotes the integer part of real number .
According to the definitions of fractional calculus, we can obtain that the fractional integral and the Caputo fractional derivative satisfy the following Lemma.
Lemma 2.3 (see ). Assume that and and . Then, for , (a), (b), (c).
Definition 2.4. Let be a real Banach space over . A nonempty convex closed set is said to be a cone, provided that(a), (b).
Definition 2.5. Let be a real Banach space and a cone. A function is called a nonnegative continuous concave functional if is continuous and for all and .
Lemma 2.6 (see ). Let be a Banach space, a cone in , and , two bounded open subsets of with and . Suppose that is continuous and completely continuous such that either (i)or (ii)holds. Then, has a fixed point in .
Let be constants, , .
Lemma 2.7 (see ). Let be a cone in real Banach space . Let be a completely continuous map and a nonnegative continuous concave functional on such that , for all . Suppose that there exist constants with such that Then, has at least three fixed points , , and satisfying
Lemma 2.8. Assume that , then be a solution of fractional boundary value problem (1.1) if and only if is a solution of integral equation where
Proof. Firstly, we prove the necessity. Let is a solution of fractional boundary value problem (1.1). By Lemma 2.3, we have
By the boundary value condition , we have
Hence, we obtain
The necessity is proved.
Now, we prove the sufficiency. Let be a solution of integral equation (2.6). Then, we have By direct computation, we obtain that That is to say, is a solution of fractional boundary value problem (1.1). Thus, the sufficiency is proved.
Lemma 2.9. Let . Then, the solution of fractional boundary value problem (1.1) satisfies (1), (2).
Proof. Suppose that is a solution of fractional boundary value problem (1.1). By (2.11), we know that
which implies that is concave on . The statement is proved.
Since , we know that is nonincreasing. By , we have , . Thus, is increasing. Noting , we obtain that for . The statement is proved.
Lemma 2.10. The Green's function , defined by (2.7), satisfies (1), (2), (3).
Proof. By (2.7), we have
It is clear that . Therefore, is increasing respect to for . Thus, . The statement holds.
If , then, Since is increasing respect to for , it is easy to see that for . We get the statement .
On the other hand, we have If , then If , then Thus, This yields the statement (3). The proof is finished.
3. Main Results
In this section, we establish the results for the existence and multiplicity of monotone and concave positive solutions for fractional boundary value problem (1.1).
Let with . We define the cone by
And denote the operator by
Lemma 3.1. Assume that , then is completely continuous.
Proof. In view of non-negativeness and continuity of and , we know that the operator is continuous and , for .
By (2.16), we have
Moreover, it follows from Lemma 2.10 that for , Therefore, the operator is well defined.
Assume that is bounded; that is, there exists a positive constant such that for all . Let . For all , we have which shows that is uniformly bounded.
In addition, for each , such that , we have Thus, by the standard arguments, we obtain that is equicontinuous. The Arzela-Ascoli theorem implies that is completely continuous. The proof is completed.
Theorem 3.2. Let . Assume that there exist two positive constants such that , . Then, fractional boundary value problem (1.1) has at least one positive, increasing, and concave solution such that .
Proof. Lemmas 2.8 and 3.1 imply that is completely continuous and fractional boundary problem (1.1) has a solution if and only if satisfies the operator equation .
Let . By , for and , we have So,
Let . For , and , it follows from that Lemma 2.6 implies that the fractional boundary value problem (1.1) has at least one positive solution such that . By Lemma 2.9, the solution is also increasing and concave.
In order to use Lemma 2.7, we define the nonnegative continuous concave functional by .
Theorem 3.3. Suppose that and there exist constants such that the following conditions hold: ,,.Then, the fractional boundary problem (1.1) has at least three positive, increasing, and concave solutions , , and such that
Proof. By Lemmas 2.8 and 3.1, is completely continuous, and fractional boundary value problem (1.1) has a solution if and only if satisfies the operator equation .
First of all, we will prove the following assertions.
Assertion 3.4 ( and ). Firstly, Lemma 3.1 guarantees . Secondly, for all , we have . By , which implies that . In the same way, .Assertion 3.5 (, and , for all ). Let , . Then, and . Consequently, .
If , then from and Lemma 2.10, we obtain that That is, , for all .Assertion 3.6 (, for all with ). If and , similar to the above, we also have .
Assertions imply that all conditions of Lemma 2.7 hold. Therefore, the fractional boundary value problem (1.1) has at least three positive solutions , and satisfying By Lemma 2.9, the positive solutions are also increasing and concave. The proof is completed.
In this section, we will present some examples to show the effectiveness of our work.
Example 4.1. Consider the fractional boundary value problem Setting , we obtain Let . We have Theorem 3.2 implies that fractional boundary value problem (4.1) has at least one positive, increasing, and concave solution. The approximate solution is obtained by the Adams-type predictor-corrector method , which is displayed in Figure 1 for the step size .
Example 4.2. Consider the fractional boundary value problem
Setting , we know that . Choosing , we obtain that Theorem 3.3 implies that fractional boundary value problem (4.4) has three positive, increasing, and concave solutions such that For numerical simulation case 1, Figure 2 depicts the phase responses state variables of with the step size .
This work was jointly supported by Natural Science Foundation of China (no. 10871214), Natural Science Foundation of Hunan Provincial under Grants nos. 11JJ3005, 10JJ6007, and 2010GK3008.
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